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Mathematica 6.0.1: problems with Beta Negative Binomial Distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg84662] Mathematica 6.0.1: problems with Beta Negative Binomial Distribution
• From: "Nasser Abbasi" <nma at 12000.org>
• Date: Tue, 8 Jan 2008 03:38:50 -0500 (EST)

When alpha is less than 1, the mean does not work. These 2 commands below 
should give the same answer:

Mean[BetaNegativeBinomialDistribution[0.3, 3, 10]]
Out[24]= -42.85714285714286


In[25]:= ExpectedValue[x, BetaNegativeBinomialDistribution[0.3, 3, 10], x]
During evaluation of In[25]:= Sum::div:Sum does not converge. >>

In addition, the mean given above as -42 is wrong, as clearly can be seen by 
looking at the PDF. The mean must be positive

ListPlot[Table[PDF[BetaNegativeBinomialDistribution[0.3, 3, 10], k], {k, 0, 
100}],
  AxesOrigin -> {0, 0}]

A rough estimate of the expected value can be found:
tbl = Table[{k, PDF[BetaNegativeBinomialDistribution[0.3, 3, 10], k]}, {k, 
1000000}];
mean = N[Sum[tbl[[k,1]]*tbl[[k,2]], {k, 1, 1000000}]]

Which gives about 20,000  and not -42

The Variance is also wrong:

In[105]:= Variance[BetaNegativeBinomialDistribution[0.3, 3, 10]]
Out[105]= -770.3481392557023

(Variance is negative??) Variance is the average of squared quantities, so 
it can not be negative.

I think the formulas for the mean and variance shown in help are wrong.

Nasser 

• Follow-Ups:
  • Re: Mathematica 6.0.1: problems with Beta Negative Binomial
    • From: Darren Glosemeyer <darreng@wolfram.com>